The properties of radiation from a parallel-plate waveguide with an arbitrarily flanged surface of finite size are investigated by means of the boundary element method (BEM) based on guided-mode extracted integral equations (GMEIEs). The reflection coefficient, the reflected and radiated powers, as well as the radiation pattern are numerically calculated for the incidence of a transverse magnetic guided-mode wave. For a number of cases of flanged parallel-plate waveguide, a variety of interesting phenomena have been presented numerically. These effects may be important in the design of antenna systems.
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 Over the last decade, remarkable progress has been made in the development of communication systems. Among them, noise figure, gain, output power, and efficiency at millimeter-wave frequencies have been improved significantly. However, the demand of wireless broadband communication at millimeter-wave frequency recently increased rapidly due to activities of digital multimedia-contents circulation. One of the most important problems of millimeter-wave communication is the large transmission loss in free space. For instance, the transmission loss of a signal at 60 GHz frequency for 5 meters distance between transmitter and receiver is about 82 dB [Shiomi and Yamamoto, 2002]. Therefore antennas with high output radiation power are required to compensate for the large transmission loss.
 Parallel-plate waveguide (PPW) with a flange surface is known as a fundamental structure widely used for electromagnetic wave radiation (as, e.g., feed horns, flush-mounted antennas). Although exact closed-form solutions are available only in few cases, the waveguide radiation behavior has been well understood using a number of numerical techniques and approximate theories [Rudduck and Wu, 1969; Wu et al., 1969; Lee, 1970; Hongo, 1972; Itoh and Mittra, 1974; Hongo et al., 1975; Lee and Grun, 1982; Leong et al., 1988; Butler et al., 1991; Kim et al., 1993; Park and Eom, 1993; Lee et al., 1996]. However, since most of the studies were based on approximate solutions, the presented results have been restricted to the problem of perpendicularly flanged PPW. Moreover, all the previous considerations were based upon the basic assumptions of infinite flange surface, which is not infinite in practice. It is therefore desirable to discuss the radiation properties of a PPW with an arbitrarily flanged surface of finite size, which is expected to enhance the output radiation power. Unfortunately, before this paper we could not find a method that can solve this problem accurately.
 Accordingly, in this paper the radiation properties of a PPW with an arbitrarily flanged surface of finite size are carried out by using the BEM based on the GMEIEs. The GMEIEs used here are derived for the problem of a dielectric-filled PPW with a tilted flange surface of finite size, as shown in Figure 1a, but it is possible to apply them to other problems, for example, a dielectric-filled PPW with a tapered flange surface of finite size, as shown in Figure 1b.
 The numerical results of computer simulations are presented. The reflection coefficient, the reflected and radiated powers as well as the radiation pattern are calculated numerically for an incident TM0 (i.e., TEM) guided-mode wave. The results are compared with those reported in the literature, and are confirmed by the law of energy conservation.
2. Reflection Coefficient
 Consider a PPW of wall thickness w and width d with a tilted flange surface of finite-length l radiating into free-space, as shown in Figure 1a, where the waveguide is filled by a dielectric with refractive index n1, and is assumed to satisfy the single-mode condition. Referring to Figure 1c, we denote an actual boundary (solid line) of the waveguide by Ci (i = 1–7) (notice that the boundaries C5 and C7 include the entire exterior boundary of the upper and lower walls, respectively). The boundary C0 (dotted line) does not express an actual boundary, but rather expresses a virtual boundary. The whole space is assumed to be magnetically homogeneous with a magnetic permeability μ0 = 4π × 10−7 H/m. In the following analysis, a harmonic time dependence exp(jωt) is assumed and suppressed from all electromagnetic field quantities, free-space wave number is denoted by k0 = ω/c, where c is the velocity of light in vacuum. Since the waveguide is assumed to extend infinitely in the y-direction, all field quantities are independent of y (i.e., ∂/∂y ≡ 0) and thus the electromagnetic field can be decomposed in terms of TM modes. In order to formulate the reflection coefficient, we assume that a dominant TM0 guided-mode wave is incident upon the aperture in the flange surface from inside of the waveguide. Since the magnetic fields have only a y-component under the above-mentioned condition, we denote the magnetic fields of the y-component by
in the coordinate systems (x, y, z) and (r, θ, z), as shown in Figure 1c. The incident guided-mode wave, H (x), the reflected guided-mode wave, H (x), and the radiation wave, H (x), are used to express the magnetic field quantities.
 We first consider the case in which the observation point x is in the region surrounded by the boundary C = C1 + C2 + C3 + C4 + C6. From Maxwell's equations and Green's theorem, the well-known boundary integral equation (BIE) for the magnetic field Hy (x) is given by
where ∂/∂n′ denotes the derivative with respect to the unit normal vector n to the boundary C. The boundary condition of perfect electric conductor, ∂Hy(x)/∂n = 0 on C1 + C2 + C3 + C4, is enforced in the process of deriving equation (2). In equation (2), G1 (x∣x′) represents the Green's function in free-space, whose refractive index is given by n1, and is expressed as
with H (x) denotes the zeroth-order Hankel function of second kind. As can be seen, it is difficult to solve equation (2) directly because it has the infinite-length integral boundary C1 + C3. To avoid this difficulty, we use the previously proposed idea [Tanaka and Tanaka, 2001; Chien et al., 2002, 2003a, 2003b, 2003c] that even though the magnetic fields near the aperture are very complicated, only the reflected guided-mode wave can survive at points far away from the aperture. Therefore we decompose the magnetic field on the boundary C1 + C3 into field components
and field H (x) is called the disturbed field (i.e., the total of evanescent reflected waves). In equation (4), R is the reflection coefficient. For convenience of notation, we also express the magnetic field on the boundary C2 + C4 + C6 by the same notation with the disturbed field
Using (4) and (5) in (2), we obtain an integral equation that includes the semi-infinite line integrals of the guided-mode waves along the boundary C1 + C3 as follows:
where the Green's theorem for the guided-mode waves, H (x), in the region surrounded by the boundary C1 + C0 + C3 is applied as
Because C0 is the virtual boundary, theoretically, we can obtain equation (8) with an arbitrary position of C0. To derive the expression of reflection coefficient, we move the observation point x to a point far away from the aperture. Under this condition, it is possible to approximate Green's function by asymptotic expression as
Substituting (9) into (6) and dividing both sides of resultant equation by A1 (r), we can obtain the relation
Since it is impossible for the reflected radiation field to exist at points far away from the aperture in the waveguide, we can set
Hence, if we use (14) in (12), we find that the reflection coefficient can be expressed as
Physically, the reflection coefficient is an invariant value for a certain structure of the waveguide, and thus we can use equation (15) to verify the independence of numerical results on location of the virtual boundary C0.
Since H (x) will vanish at points far away from the aperture, the integral boundary C1 + C3, which has infinite-length, can be regarded as finite-length in equation (16).
 When the observation point x is in the free-space region surrounded by the boundary C = C5 + C6 + C7 the well-known BIE for the magnetic field is given by
As can be seen, equation (19) has the integral boundary C5 + C7 also with infinite-length, but it is not difficult to truncate the boundary C5 + C7 in the numerical solution procedure at points where the magnetic field becomes small enough to be regarded as vanished.
 The BIEs (16) and (19) are to be solved numerically for the problems shown in Figures 1a and 1b by using the conventional BEM or MM. Once the fields on all the actual boundaries have been obtained, the reflection coefficient can be evaluated from equation (15), and the fields at any point can also be calculated by the boundary integral representations similar to equations (16) and (19).
4. Radiation Pattern
 The radiation field H (r, θ) in the free-space region can be expressed by using the asymptotic form of Green's function in free-space with refractive index n0 as follows:
 So far, we have discussed the case in which a dielectric material with refractive index n1 fills the inside of the waveguide. For the case of unfilled PPW, only one GMEIE is required, which is easy to derive by using the same procedure as that used in the above derivation of (16).
5. Accuracy and Convergence Tests
 We first consider the problem of an unfilled PPW with a perpendicular flange surface. If we stretch the length of flange surface l to where the magnetic field becomes small enough, the problem will become close to the conventional one. Since many papers have reported solutions to this problem before, we can compare our results with those obtained by the methods appearing in the previously published papers [Lee, 1970; Hongo, 1972; Kim et al., 1993]. In Table 1 the results of comparison for the reflection coefficient of an incident TM0 guided-mode wave are presented. As can be seen, our results are in reasonable agreement with the results reported in the literature. Notice that owing to the different exp (jωt) time convention used, there is a minus sign difference in the phase of reflection coefficient in the literature. These numerical results show the validity of the present method.
Table 1. Comparison Between the Various Methods Used to Calculate the Reflection Coefficient R of an Unfilled PPW With a Perpendicular Flange Surface of Infinite Size for k0d = 2.0
 From the above calculation, we found that the magnetic field on the flange surface decays slowly. As typically shown in Figure 2, the magnetic field becomes small enough at a rather long length of flange surface, which is approximately 160λ. It is obvious that the results in Table 1 do not reflect the practical problem, of which the flange-length is comparable with the waveguide width. Consequently, we predict that the radiation properties of a PPW with a finite flange surface are different from those of a PPW with an infinite flange surface.
 We next apply the method to the problem of a filled PPW with a tilted flange surface of finite size, as shown in Figure 1a. Since the problem seems to be difficult to solve using the methods based on approximate theories, no one, to our knowledge, has reported solutions of this kind of problem before. In Table 2 the results of reflected power ΓR, radiated power ΓS, and their total ΓTOTAL, are presented for the case of d = 0.3124λ, w = 0.079λ, l = 25λ, and n1 = 1.6. These results, which satisfy the energy conservation law within an accuracy of 1%, verify the feasibility of the method in this paper.
Table 2. Reflected Power ΓR, Radiated Power ΓS, and Their Total ΓTOTAL of a Dielectric Filled PPW With a Tilted Flange Surface for d = 0.3124λ, w = 0.079λ, l = 25λ, and n1 = 1.6
 In section 2 it has been stated that the reflection coefficient is independent of location of the virtual boundary C0. For numerical demonstration, the reflection coefficient of a PPW, as considered in Table 2, as a function of location of C0 is plotted in Figure 3 for tilting angle ϕ = 0°. As can be observed, the reflection coefficient is independent of location of the virtual boundary C0 except at ∣a/λ∣ < 0.1. This error is caused by the numerical method used, because when C0 approaches the aperture the boundaries C2 and C4 approach zero.
 In order to verify the truncation of the semi-infinite boundaries in the numerical solution procedure, we explore the distributions of disturbed field on C1 (solid curve) and total field on longitudinal part of C5 (dotted curve) in Figure 4, where the waveguide parameters are the same as in Table 2 with tilting angle ϕ = 0°. It is seen that both the disturbed and total fields can be regarded as having vanished at approximate boundary-length 16λ. The results in Figure 4 show us that using the BEM based on the GMEIEs certainly can treat the waveguide discontinuity problem as an isolated object of finite size, and thus it is suitable for the basic theory of computer-aided design (CAD) software for waveguide circuits.
6. Numerical Simulations
 In the first sequence of calculations we study the effect of size of the flange surface on the radiation properties of a PPW as considered in Table 2. With tilting angle ϕ = 0° and by changing size of the flange surface, including l and w, the calculations are carried out for both radiated power and radiation pattern. Notice that the maximum of wall thickness w is 0.5d (i.e., 0.1562λ), which agrees with practice. It is out of our prediction the radiated power ΓS almost does not depend on the flange-size l and w. However, on the contrary, the radiation pattern strongly depends on the flange-length l (not on the wall thickness w), as shown in Figure 5 for the case of l = 160λ (solid curve), 25λ (dotted curve), and 5λ (dashed curve). As can be observed, the radiation pattern fluctuates due to the interference between two diffracted waves from the far edges of the flange surface. Amplitude of the fluctuation is inversely proportional to the flange-length, and theoretically equals zero with an infinite-length flange surface. In fact, this phenomenon is not difficult to imagine, but as far as we know, no one has reported numerical solutions to this potentially important problem.
 In subsequent calculations we apply the method to a number of cases of flanged PPW, as shown in Figures 1a and 1b. With the same waveguide parameters as in Table 2 the calculated results are shown below.
 For the system shown in Figure 1a the dependence of radiated power ΓS on tilting angle ϕ and the typical radiation patterns are respectively shown in Figures 6a and 6b. The results in Figure 6a show that the radiated power of a flanged PPW can be improved significantly by using a tilted flange surface. From Figure 6b the radiation pattern becomes asymmetric with increasing the tilting angle, which is also observed in accordance with the result in Figure 6a where the far-field intensity is greatly enhanced.
 For the system shown in Figure 1b that we call a dielectric-filled PPW with a tapered flange surface, the dependence of radiated power ΓS on tapering angle ϕ and the typical radiation patterns are shown in Figures 7a and 7b, respectively. It should be noted that the tapering angle ϕ is anticlockwise with respect to the x axis, as shown in Figure 1c. Figure 7a shows us that the radiated power decreases with increasing the tapering angle (i.e., ϕ > 0°), but on the contrary, it increases with decreasing the tapering angle (i.e., ϕ < 0°). Accordingly, the far-field intensity is greatly enhanced, and the beam width is sufficiently reduced by using an uptapered flange surface.
 In summary, it is seen that a tilted or uptapered flange surface has improved the radiated power of a flanged PPW as it has matched the waveguide flange. These results are interesting, and may be important for millimeter-wave free-space communication systems.
 The radiation properties of a dielectric filled and unfilled PPW with an arbitrarily flanged surface of finite size have been studied by the BEM based on the GMEIEs. On the basis of the theory developed in sections 2, 3, and 4, typical numerical evaluations have been performed for the case of an incident TM0 (i.e., TEM) guided-mode wave. The numerical results were confirmed by using the law of energy conservation. It has been found that the numerical results are in reasonable agreement with previously published results and physical consideration. Although the analysis presented here is applicable to a more general class of flanged PPW, the two types of flange surface are interesting from the practical viewpoint.
 We have shown that it is possible to enhance the radiated power of a finitely flanged PPW by using a tilted or uptapered flange surface. In addition, the fluctuation of radiation pattern of a flanged PPW with a finite-length flange surface, which may be important for millimeter-wave free-space communication systems, has also been presented. Since we do not employ any approximation in the derivation of the GMEIEs, adapting the GMEIEs to more complicated waveguide circuits having more than one port, and more complicated end-shapes, is straightforward.